Advanced Sudoku Strategy

Now that you have mastered the basic strategies you can tackle almost any Sudoku puzzle. With the full range of tactics even puzzles categorised as 'fiendish' can be solved. And yet one of the delights of Sudoku is that there is another level of advanced strategies to call upon that distinguish the Sudoku Masters from the rest. It is amazing how such an apparently simple puzzle can lead to such complexity.

Colorful Puzzles

When describing the X-Wing and Swordfish the strategy employed more complex logic. The tactics involved looking beyond one or two groups of squares to several interlocking groups. Both these strategies rely on pairs of squares. They investigate what would happen with different allocation options and finds the squares that are constrained by both alternate allocations. This pair rule only applies for squares where one number can only go in one of two squares in a single group (this may be a row; column or region), if there are three or more it can't be used.

Here is an example of an X-Wing. There are two rows where there are only two squares which can take a 4 (rows A and H) These match up by columns to form a box. There are only two alternative allocations for 4 in Ac and Hh or Ah and Hc. This is conveniently illustrated by using alternate blue and orange coloring for the squares. Either the blue squares take a 4 or the orange squares take a 4. This is useful because it's now unquestionable that a 4 occurs in both column c and h so all the squares that also looked as though they could have taken a 4 in these two columns can be discounted (these are shown in pink). Using alternate colors helps to show what is going on. Sudoku Dragon supports six different colors for marking squares.

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So now let's turn to the Swordfish to see if coloring helps there too. In this Swordfish there are three rows (A, B and H) all with a pair possible squares for a 6. As the squares are all located in just three columns b; g and i they form a 'Swordfish'. The 6s must either go on the blue squares or the orange squares they can not be allocated any other way. Because the three columns b; g and i all contain both a blue and orange square then a 6 can not go in any other square in these columns. So the pink square Gi can not actually take a 6 even though it looks as though it could do. Once again coloring the pairs has made it easy to see the options.

Alternate Pair Exclusion

The X-Wing and Swordfish are examples of a more general rule. This is always satisfying for a mathematician, by looking at simple examples it is possible to extend the same reasoning to a general rule.

If you identify pairs of squares in the same row; column or region for the same number you can start coloring them. If you find another linked pair sharing any square with a colored pair then you can continue using the same alternate coloring scheme for that pair too. These are termed 'alternate pairs' as the true allocation must be in one of the alternative squares in the pair, you will sometimes see these called 'conjugate pairs', but as this term comes from mathematics, it is not very descriptive.

When you have finished coloring the linked pairs you can then look at the pattern and make use of the color combinations in groups. If you find a group (row; column or region) that has both colors in any of its squares then any other square that could that number can be safely excluded as a possibility. This general rule covers not only the X-Wing and Swordfish types but a whole range of other patterns of possibilities too.

Here is a grid with the possibilities for 6 colored. There are pairs of possibilities for 6 in row B and C as well as region Ag (it is often a region pair that proves helpful). There is also a pair in column b that does not interlink with another pair and so does not help. The coloring shows that column a has both an orange and a blue square in it so the 'possible' 6 in square Ha (pink) can be knocked out of the possibilities. For more on Sudoku Dragon's support see our square coloring page.

Sudoku Dragon highlights alternate pairs and uses the alternate pair strategy to solve the more difficult puzzles.

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However the hard work of alternate pair coloring does may not help solve any squares. It is a technique to hold in reserve for very tricky puzzles because it takes so long to work out. Pairs are most frequently found whether you have a dozen or so possible squares left for a number, they are not as common at the beginning or end of Sudoku puzzle solving.

Alternate Pair Deduction

Just when you thought solution strategies were becoming too mind blowingly complex, there is yet further twists. This is another excuse to get out your crayons and start coloring Sudoku squares. It is slightly different to the previous section on 'alternate pair exclusion', the pairs are identified and colored in just the same way but in this case the logic is different and often more useful. If after coloring you have any group with two squares of the same color then something is distinctly odd as that implies you could have two squares with the same symbol in the same group. If it produces two or more squares of the same color then this particular color assignment is not possible and the other color must be the correct one and all those squares can all be set as the only possible squares for the symbol.

Starting with orange for the square Fd and blue for its alternate row pair Fe, the pairs in column e come into play and orange goes into Ae. From Fd there is a pair in column d so blue for Id, then following the pair in row I orange goes in Ia, now there is a pair in region Ga which means Gb must be blue, finally column b is a pair so Ab is orange too. So finally we end up with two orange squares in row A, which can not be right. If the 7s went in the orange squares there would be two 7s in row A. So the 7s can not go in the orange squares they must go in the blue squares Fe; Gb and Id.

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For an extensive run-through example of this strategy  click here

Multicolored Alternate Pairs

Once you have mastered Alternate Pairs it is worth introducing a further twist that is rarely needed. When you color a grid you will quite often find there are two independent chains of interlinked pairs. You can use two more colors to mark up the other set of pairs and you end up with more than just a very colorful grid! The network of linked colored squares can restrict and force square allocations rather like in the case of a single network of colored pairs.

Here's a puzzle with all the squares that can take a 6 have been marked using Sudoku Dragon. There are some useful looking pairs. In region Dg the orange square Eh and blue square Fg which links on Row F to orange pair Ff, and Eh links through column h to blue square Gh. Square Fg links through column g to orange Hg. That's it for that set but there is another pair in column c. Square Gc has been colored red and Ic green. Now the crucial fact is that the two sets are related via row G (NOTE: the possibility in Gd forbids the two sets being fully interlinked). If a 6 were to go in the blue squares it could not go in the red square it would have to go in the green square Ic. Alternatively if a 6 was correct for the orange squares then this does not intersect with the red-green pair directly so it can't be deduced which one is correct. However, putting the two alternatives together where orange and green squares intersect it's now where certain that a 6 can't go, and in this example it is square If marked in pink. Either there is a 6 is in orange square Ff or a '6' in green square Ic so both possible cases forbid a 6 in the pink square If which intersects this row and column.

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Further twists

What was achieved with pairs could equally be achieved with triplets or quadruplets. All that is needed is to work through the logic and see if all the three or four alternatives all combine to have a common implication. However these are too rare and too hard to identify to be of practical use.

Hook or X-Y Wing

When I first saw the term X-Y Wing I thought someone had just mistyped X-Wing or that it was yet another variant of it. However, the X-Y Wing is another tricky advanced strategy separate from pairs but once again using the 'either-or' logic that flows from using connected groups.

To avoid confusion we'll use the term Hook for this technique. Under rather special circumstances, the Hook knocks out other possibilities. An hook (or X-Y Wing) requires you to find three squares. The squares must all have two possibilities each in three different numbers. The three squares form a chain of pairs of possibilities. An example of such a chain is [2; 5] [5;7] and [7;2] expressed this way you can see the chain of possibilities is [2 -> 5 -> 7 -> 2]. Where these squares are located is also important, two must be in the same row or column, they form the stem and the other must be in the same region as one of the other two squares.

It is all too easy to slip up on correct identification of where a Hook is located. How is this obscure relationship useful? Well the way these squares is arranged restricts possibilities elsewhere in the grid. If in our example the stem contains [2,5] and [5,7] then 5 is the stem number and the [2,7] the branch or hook. If 5 is the correct choice for the [2,5] square then that means the [5,7] square must be 7. The only alternative for the [2,5] square is 2, now if this is '2' then this forces the [2,7] square to be 7 and therefore the [5,7] square must be 5. These are the only two choices and if there are any squares where a 7 is not possible for both these two alternatives then we can safely exclude 7 as a possibility for them.

Here are some pictures of a real puzzle that may help. It is a puzzle at a stage where the easy squares have run out.

The light green squares are of interest, they are Bd that can be only [2,7]; Ce that can take [2,5] (the 9 is not possible because of the naked twin [4,9] in row C) and Ch that can take [5,7] ) (the 9 is excluded for the same reason). So we have our chain [2,7] [7,5] [5,2] of three squares. The last two form the stem of the hook and 7 is the hook number. Following our logic if the 2 went in Ce this means a 7 must go in Bd and a 5 in Ch. So this gives us the following segment of the grid.

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The green squares are the ones we have tentatively allocated and the blue squares indicate where another 7 can not be put in this scenario.

The only other option for Ce was for it to be a 5 rather than a 2, and this forces us to put a 7 in Ch. So now we have the alternative scenario with Ch and Ce highlighted in green and for this alternative the blue squares show where a 7 can not be put.

Well, we are in luck as there is one square which is blue in both the alternatives, this is square Bh - which we have coloured darker blue to highlight it. In general there could be more than one square in common and the rule would apply to all the squares common to the two alternatives. Square Bh can not be a 7 for either of the two possibilities for Ce (5 or 2) and so 7 can be safely excluded from the possibilities for this square. In the case of this particular puzzle, this is crucial, as that leaves only one choice for Bh as it can not be a 7 it must be the remaining possibility of '9', the hook strategy enables us to immediately solve a square and it turns out to be the last tricky one to solve.

The Hook is a general technique, the term X-Y Wing name comes from its mathematical formulation as three squares containing [x, y]; [y, z] and [x, z]. (In our case x=5; y=2; z=7). It tells us that the shared squares where the two alternative allocations for z intersect can not possibly contain a z.

For an extensive run-through example of this strategy  click here

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Have you mastered all the strategies for solving Sudoku? Our strategy page gives an introduction to all the main tactics for solving a Sudoku puzzle. We have separate pages on Trial and error and Advanced Strategies.. Read more...

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